Definition 4.1 ([1]}, [2]}) 
A random walk on a strongly connected directed graph \(\mathcal {G}=(\mathcal {V},\mathcal {E},w)\)  is a homogeneous Markov chain \(\mathcal {X}=(X_{n})_{n\ge 0}\)  not necessarily reversible with a finite state space \(\mathcal {V}\)  and whose transition probabilities are proportional to the edges weights. In particular, the entries of \(\mathbf {P}\) , written as \(p(x,y), \forall {x,y \in \mathcal {V}}\) , give the transition matrix associated to \(\mathcal {X}\)  on \(\mathcal {V}\) , and they are :
\(p(x,y)=\mathbb {P}(X_{n+1}=y|X_n=x).\) 

